Question

Let $f:R\to R$ be a positive increasing function with $\underset{x\to \infty }{\lim }\frac{f\left(3x\right)}{f\left(x\right)}=1$
Then $\underset{x\to \infty }{\lim }\frac{f\left(2x\right)}{f\left(x\right)}=$

• A. $\frac{2}{3}$
• B. $\frac{3}{2}$
• C. $3$
• D. $1$

Answer 1

Answer: (D)

f(x) is a positive increasing function.
$\therefore 0<f(x)<f(2x)<f(3x)$
$\Rightarrow 0<1<\frac{f(2x)}{f(x)}<\frac{f(3x)}{f(x)}$
$\Rightarrow \underset{x\to \infty }{\lim }1\le \underset{x\to \infty }{\lim }\frac{f\left(2x\right)}{f\left(x\right)}\le \underset{x\to \infty }{\lim }\frac{f\left(3x\right)}{f\left(x\right)}$
By Sandwich Theorem.
$\Rightarrow \underset{x\to \infty }{\lim }\frac{f\left(2x\right)}{f\left(x\right)}=1$